I noticed in that shops, potential crew members' experience bars are shown, but I have only seen them empty. Can crew members in shops start with some amount of experience? What about crew you get elsewhere?

As of the Advanced Edition, at least, it is possible - however generally all EXP bars will be empty until you reach later sectors, and even then they will have a "speciality", ie. a single stat with EXP with the others all empty (which is how my crew generally ends up anyway).

This can apply to new crew members received from events too, as will be shown by the corresponding icon next to the crew member in the message telling you about them joining you.